On the noise performances of fractal-fractional electrical circuits

In this work, the noise performances of the fractal-fractional electrical circuits have been addressed. The nonlocal fractal calculus has been adopted as our mathematical basis. The fractal time component has also been included for the physical measurability of electrical quantities. The derivations of crucial stochastic parameters of circuit responses, which determine their noise performances, have been performed. Numerical simulations have also been conducted where the influences of Hausdorff dimension of the fractal set, orders of fractal-fractional reactive components, and other parameters on the noise performances have been studied. Regardless to any specific circuit, we have found that the noise performances can be improved by increasing the orders of fractal-fractional reactive components. The optimum Hausdorff dimensions, which the best noise performances can be achieved given the orders of fractal-fractional reactive components, have also been calculated. The results proposed in this work serve as the foundation for understanding noise in fractal-fractional electrical circuits and can be extensively applied to large-scaled circuits, for example, the infinite circuit networks and so forth.